This invention relates to a method of convolutive coding for the transmission of space-time block codes according to the technique termed Golden Code. It also relates to a system implementing such a method.
In a general fashion, the invention is applicable to the field of transmission or radio broadcasting of digital data, or sampled analogue data, in particular in the case of transmission with mobiles or, in an even more general manner, in the case of local wireless networks or not. The invention can in particular, if it is wished, be applied to high data rate wireless transmissions. A first application category relates to cellular communication with mobiles, such as the UMTS, for example. A second application category relates to local wireless networks. A third category is that of future ad hoc networks.
In a more precise manner, this invention is applicable to multi-antenna MIMO (“Multiple Input Multiple Output”) systems implementing space-time block codes of the Golden Code type.
A Space-Time Block Code (STBC) is a finite set Γ of complex matrices (the codewords) having M lines and T columns and in which each component Γit is the symbol which will be transmitted over the antenna i (1≦i≦M) at the instant t (1≦t≦T). An STBC is square if M=T.
The construction criteria for an STBC as described in particular in the document by V. Tarokh, N. Seshadri and A. R. Calderbank, “Space Time Codes for High Data Rates Wireless Communication: Performance Criterion and Code Construction”, IEEE Transactions on Information Theory, vol. 44, no. 2, March 1998, are as follows:                The order of diversity (of transmission), marked d, of an STBC is defined by:        
  d  =            Min                        X          ⁢                                          ⁢          Y          ⁢                                          ⁢          ɛ                          X          ≠          Y                      ⁢                  ⁢          rank      (              X        -        Y            )      
An STBC is of full diversity if the order of diversity is maximum, i.e. d=Min (M, T).                The coding gain, marked g, of an STBC of full diversity with T≧M is defined by:        
  g  =            Min                        X          ,                      Y            ⁢                                                  ⁢            ɛΓ                                    X          ≠          Y                      ⁢                  Det        ⁡                  (                                    (                              X                -                Y                            )                        ⁢                                          (                                  X                  -                  Y                                )                            H                                )                    M      
It has been demonstrated that the average error probability by codeword, on a Rayleigh fading channel, is proportional to 1/(gSNR)dN at high Signal to Noise Ratio. In order to maximise the performances of an STBC over this type of channel, it is therefore necessary to maximise the coding gain.
As indicated previously, this invention relates more particularly to a space-time block code of the Golden Code type.
The Golden Code, as defined in the document by J.-C. Belfiore, G. Rekaya and E. Viterbo, “The Golden Code: A 2×2 Full-Rate Space Time Code with Non-Vanishing Determinants”, IEEE Transactions on Information Theory, vol. 44, no. 2, April 2005 is a square STBC with 2 transmit antennas (M=2 and T=2). This code is with full diversity (d=2). It offers the best coding gain as of today. This is the algebraic construction of the Golden Code which can carry out its partitioning.
In order to define the Golden Code, it is necessary to introduce division cyclic algebra (non-switching body), A0, constructed on the body Q[i, θ] in the following manner:
            A      Q        =          {                                    [                                                                                a                    +                                          b                      ·                      θ                                                                                                            c                    +                                          d                      ·                      θ                                                                                                                                        i                    ⁡                                          (                                              c                        +                                                  d                          ·                                                      θ                            _                                                                                              )                                                                                                            a                    +                                          b                      ·                                              θ                        _                                                                                                                  ]                    ⁢                                          ⁢          with          ⁢                                          ⁢          a                ,        b        ,        c        ,                  d          ∈                      Q            ⁡                          [              i              ]                                          }        where                                                        i              2                        =                          -              1                                ,          θ                                              =                                    1              +                              5                                      2                                and            θ      _        =                  1        -                  5                    2      
The ring, marked A2, is defined on this algebra while restricting a, b, c and d in the ring of Gauss integers Z[i]:
      A    Z    =      {                            [                                                                      a                  +                                      b                    ·                    θ                                                                                                c                  +                                      d                    ·                    θ                                                                                                                        i                  ⁡                                      (                                          c                      +                                              d                        ·                                                  θ                          _                                                                                      )                                                                                                a                  +                                      b                    ·                                          θ                      _                                                                                                    ]                ⁢                                  ⁢        with        ⁢                                  ⁢        a            ,      b      ,      c      ,              d        ∈                  Z          [          i          ]                      }  
The infinite Golden Code, marked Γ∞, is a principal ideal of the ring AZ generated by an element a of this ring. It should be noted that if a vectorial instead of a matricial representation is chosen, AZ and therefore the Golden Code are real 8-dimensional lattices. The element a was chosen such that the ideal forms a lattice corresponding to √{square root over (5)}Z8. This is why the Golden Code includes a normalisation by 1/√{square root over (5)}. Accordingly:
            Γ      ∞        =          {                                    1                          5                                ⁢                      α            ⁡                          [                                                                                          a                      +                                              b                        ·                        θ                                                                                                                        c                      +                                              d                        ·                        θ                                                                                                                                                        i                      ⁡                                              (                                                  c                          +                                                      d                            ·                                                          θ                              _                                                                                                      )                                                                                                                        a                      +                                              b                        ·                                                  θ                          _                                                                                                                                ]                                ⁢                                          ⁢          with          ⁢                                          ⁢          a                ,        b        ,        c        ,                  d          ∈                      Z            [            i            ]                              }        where      α    =          [                                                  1              +              i              -                              i                ·                θ                                                          0                                                0                                              1              +              i              -                              i                ·                                  θ                  _                                                                        ]      
It has been shown that the infinite Golden Code coding gain is 1/√{square root over (5)}. If a finite code is used, it suffices to constrain a, b, c and d to belong to a finite sub-set included in Z[i] (a QAM constellation for example). The fact that the infinite Golden Code corresponds to Z8 greatly facilitates binary labelling and “shaping” (the fact of extracting a constellation) while guaranteeing a good Euclidean distance between codewords.